Cartan-Weyl 3-algebras and the BLG Theory II: Strong-Semisimplicity and Generalized Cartan-Weyl 3-algebras
arXiv:1004.1513 · doi:10.1007/JHEP02(2011)037
Abstract
One of the most important questions in the Bagger-Lambert-Gustavsson (BLG) theory of multiple M2-branes is the choice of the Lie 3-algebra. The Lie 3-algebra should be chosen such that the corresponding BLG model is unitary and admits fuzzy 3-sphere as a solution. In this paper we propose another new condition: the Lie 3-algebras of use must be connected to the semisimple Lie algebras describing the gauge symmetry of D-branes via a certain reduction condition. We show that this reduction condition leads to a natural generalization of the Cartan-Weyl 3-algebras introduced in arXiv:1004.1397. Similar to a Cartan-Weyl 3-algebra, a generalized Cartan-Weyl 3-algebra processes a set of step generators characterized by non-degenerate roots. However, its Cartan subalgebra is non-abelian in general. We give reasons why having a non-abelian Cartan subalgebra may be just right to allow for fuzzy 3-sphere solution in the corresponding BLG models. We propose that generalized Cartan-Weyl 3-algebras is the right class of metric Lie 3-algebras to be used in the BLG theory.
arXiv comment28 pages. LaTeX. v2. typos corrected
References in corpus (36)
- Gauge Symmetry and Supersymmetry of Multiple M2-Branes
- Algebraic structures on parallel M2-branes
- Modeling Multiple M2's
- Comments on Multiple M2-branes
- On Foundation of the Generalized Nambu Mechanics
- The M2-M5 Brane System and a Generalized Nahm's Equation
- The Noncommutative Bion Core
- On the Quantization of Nambu Brackets
- M2 to D2 revisited
- Deformation Quantization and Nambu Mechanics
- Classical and Quantum Nambu Mechanics
- Bagger-Lambert Theory for General Lie Algebras
- Boundary Conditions for Interacting Membranes
- N=8 superconformal gauge theories and M2 branes
- Towards the Quantum Geometry of the M5-brane in a Constant C-Field from Multiple Membranes
- M2-Branes and Background Fields
- Deformation Quantization of Superintegrable Systems and Nambu Mechanics
- Light-cone M5 and multiple M2-branes
- Metric Lie 3-algebras in Bagger-Lambert theory
- Multiple Self-Dual Strings on M5-Branes
- Quantized Nambu-Poisson Manifolds and n-Lie Algebras
- Truncated Nambu-Poisson Bracket and Entropy Formula for Multiple Membranes
- SDiff Gauge Theory and the M2 Condensate
- Metric 3-Lie algebras for unitary Bagger-Lambert theories
- Hamilton-Jacobi equations and Brane associated Lagrangians
- Volume Preserving Diffeomorphism and Noncommutative Branes
- M-theory and Deformation Quantization
- Comments on the topological open membrane
- Cubic Matrix, Nambu Mechanics and Beyond
- A Concise Review on M5-brane in Large C-Field Background
- On M-Algebras, the Quantisation of Nambu-Mechanics, and Volume Preserving Diffeomorphisms
- Fuzzy Funnels: Non-abelian Brane Intersections
- String solitons in the M5-brane worldvolume with a Nambu-Poisson structure and Seiberg-Witten map
- D1-brane in Constant R-R 3-form Flux and Nambu Dynamics in String Theory
- Cartan-Weyl 3-algebras and the BLG Theory I: Classification of Cartan-Weyl 3-algebras
- Structure of Cubic Matrix Mechanics
Cited by in corpus (4)
- D1-Strings in Large RR 3-Form Flux, Quantum Nambu Geometry and M5-Branes in C-Field
- 3-Leibniz bialgebras (3-Lie bialgebras)
- Cartan-Weyl 3-algebras and the BLG Theory I: Classification of Cartan-Weyl 3-algebras
- Quantum corrections of (fuzzy) spacetimes from a supersymmetric reduced model with Filippov 3-algebra